On the k–Lucas Numbers of Arithmetic Indexes

Affiliation(s)

Department of Mathematics and Institute for Applied Microelectronics (IUMA), University of Las Palmas de Gran Canaria (Spain).

Department of Mathematics and Institute for Applied Microelectronics (IUMA), University of Las Palmas de Gran Canaria (Spain).

Abstract

In this paper, we study the k–Lucas numbers of arithmetic indexes of the form an+r , where n is a natural number and r is less than r. We prove a formula for the sum of these numbers and particularly the sums of the first k-Lucas numbers, and then for the even and the odd k-Lucas numbers. Later, we find the generating function of these numbers. Below we prove these same formulas for the alternated k-Lucas numbers. Then, we prove a relation between the k–Fibonacci numbers of indexes of the form 2rn and the k–Lucas numbers of indexes multiple of 4. Finally, we find a formula for the sum of the square of the k-Fibonacci even numbers by mean of the k–Lucas numbers.

In this paper, we study the k–Lucas numbers of arithmetic indexes of the form an+r , where n is a natural number and r is less than r. We prove a formula for the sum of these numbers and particularly the sums of the first k-Lucas numbers, and then for the even and the odd k-Lucas numbers. Later, we find the generating function of these numbers. Below we prove these same formulas for the alternated k-Lucas numbers. Then, we prove a relation between the k–Fibonacci numbers of indexes of the form 2rn and the k–Lucas numbers of indexes multiple of 4. Finally, we find a formula for the sum of the square of the k-Fibonacci even numbers by mean of the k–Lucas numbers.

Cite this paper

S. Falcon, "On the k–Lucas Numbers of Arithmetic Indexes,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1202-1206. doi: 10.4236/am.2012.310175.

S. Falcon, "On the k–Lucas Numbers of Arithmetic Indexes,"

References

[1] S. Falcon, “On the k-Lucas Numbers,” International Journal of Contemporary Mathematical Sciences, Vol. 6, No. 21, 2011, pp. 1039-1050

[2] S. Falcon and A. Plaza, “On the Fibonacci k-Numbers,” Chaos, Solitons & Fractals, Vol. 32, No. 5, 2007, pp. 1615-1624. doi:10.1016/j.chaos.2006.09.022

[3] S. Falcon and A. Plaza, “The k-Fibonacci Sequence and the Pascal 2-Triangle,” Chaos, Solitons & Fractals, Vol. 33, No. 1, 2007, pp. 38-49.
doi:10.1016/j.chaos.2006.10.022

[4] S. Falcon and A. Plaza, “On k-Fibonacci Numbers of Arithmetic Indexes,” Applied Mathematics and Computation, Vol. 208, 2009, pp. 180-185
doi:10.1016/j.amc.2008.11.031